Salomaa gave an axiomatization for equivalence of regular
expressions, and we know that checking equivalence is complete for
polynomial space. Regular languages can be described more succinctly by
using additional operations like synchronized shuffle (also called merge),
renaming and hiding. Equivalence of these expressions for traces is
axiomatized using Milner’s expansion law (or Bergstra and Klop’s left and
right merge operations), and checking equivalence is complete for
exponential space. If we disallow nesting of shuffle, renaming and hiding,
checking equivalence is still in polynomial space. We give a proof system
for a fragment. We do not use the expansion law. The syntax matches
languages corresponding to 1-bounded free choice Petri nets.